The number of permutations with k inversions

TL;DR

This paper provides explicit formulas for counting permutations with a given number of inversions and for certain composition counts, using advanced algebraic tools like Gr"obner bases and free resolutions.

Contribution

It introduces new explicit formulas for $I_n(t)$ and $H(n,d,t)$ leveraging algebraic methods, advancing combinatorial enumeration techniques.

Findings

Explicit formulas for $I_n(t)$ and $H(n,d,t)$ are derived.

Uses Gr"obner bases and free resolutions for combinatorial enumeration.

Connects algebraic methods with permutation and composition counting.

Abstract

Let $n\geq 1$, $0\leq t\leq {n \choose 2}$ be arbitrary integers. Define the numbers $I_n(t)$ as the number of permutations of $[n]$ with $t$ inversions. Let $n,d\geq 1$ and $0\leq t\leq (d-1)n$ be arbitrary integers. Define {\em the polynomial coefficients} $H(n,d,t)$ as the numbers of compositions of $t$ with at most $n$ parts, no one of which is greater than $d-1$. In our article we give explicit formulas for the numbers $I_n(t)$ and $H(n,d,t)$ using the theory of Gr\"obner bases and free resolutions.